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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 41382.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41382.cd1 | 41382bu3 | \([1, -1, 1, -466115, -122368705]\) | \(8671983378625/82308\) | \(106298145592452\) | \([2]\) | \(414720\) | \(1.8538\) | |
41382.cd2 | 41382bu4 | \([1, -1, 1, -455225, -128366917]\) | \(-8078253774625/846825858\) | \(-1093648470927942402\) | \([2]\) | \(829440\) | \(2.2003\) | |
41382.cd3 | 41382bu1 | \([1, -1, 1, -8735, 26183]\) | \(57066625/32832\) | \(42401476358208\) | \([2]\) | \(138240\) | \(1.3044\) | \(\Gamma_0(N)\)-optimal |
41382.cd4 | 41382bu2 | \([1, -1, 1, 34825, 182999]\) | \(3616805375/2105352\) | \(-2718994671470088\) | \([2]\) | \(276480\) | \(1.6510\) |
Rank
sage: E.rank()
The elliptic curves in class 41382.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 41382.cd do not have complex multiplication.Modular form 41382.2.a.cd
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.